Compute
\[\sum_{n = 2}^{10000} \frac{1}{n \sqrt{n - 1} + (n - 1) \sqrt{n}}.\]
Explanation: We have that
\begin{align*}
\frac{1}{n \sqrt{n - 1} + (n - 1) \sqrt{n}} &= \frac{n \sqrt{n - 1} - (n - 1) \sqrt{n}}{(n \sqrt{n - 1} + (n - 1) \sqrt{n})(n \sqrt{n - 1} - (n - 1) \sqrt{n})} \\
&= \frac{n \sqrt{n - 1} - (n - 1) \sqrt{n}}{n^2 (n - 1) - (n - 1)^2 n} \\
&= \frac{n \sqrt{n - 1} - (n - 1) \sqrt{n}}{n(n - 1)(n - (n - 1))} \\
&= \frac{n \sqrt{n - 1} - (n - 1) \sqrt{n}}{n(n - 1)} \\
&= \frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}.
\end{align*}Thus,
\begin{align*}
\sum_{n = 2}^{10000} \frac{1}{n \sqrt{n - 1} + (n - 1) \sqrt{n}} &= \left( 1 - \frac{1}{\sqrt{2}} \right) + \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}} \right) + \left( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} \right) + \dots + \left( \frac{1}{\sqrt{9999}} - \frac{1}{\sqrt{10000}} \right) \\
&= 1 - \frac{1}{100} = \boxed{\frac{99}{100}}.
\end{align*}